Gravitation and the Red Shift

GRAV lT/; TI ON
and
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THE RED SHIFT
Julian M. Avery
Apr i I , l 9E,6
Revised July, 1966
REFERENCES
(1) Einstein "The Meaning of Relativity"
Third Edition (1950)
(2) Max Born "Einstein's Theory of Relativity" (1962)
(3) R. C. Tolman - "Relativity, Thermodynamics and
Cosmo I ogy" (1934)
(Reprinted 1946, 1949, 1958)
(4) Harlow Shapley "Galaxies"
Revised Edition (1961)
(5) Sandage "The Red Shift"
Scientific American, Sept. 1956
(6) Greenstein "Quasi-Stellar Radio Sources"
Scientific American, Dec. 1963
(7) Sir Arthur Eddington "The Expanding Universe"
1st Edition 1933, 4th Impression 1946
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TABLE OF CONTENTS
The format used in this paper is to provide first an
introduction to prepare the reader for what is to
follow; a glossary of Terms used in the Text; a Surrrnary
which gives briefly the findings which are presented;
a Text which develops the argument in a logical sequence
with the conclusions reached thereby; a set of Figures
showing graphically the results of the study, together
with a descriptive explanation; and a series of Appen­dices
which support the argument of the Text with appro­priate
detail. The reader may find it helpful to glance
at the Figures before reading the Text, in order that he
may have a clear idea as to the results the Text is lead­ing
up to as it proceeds.
i ntroduction
Glossary of Symbols Used in the Paper
The Text - Gravitation and The Red Shift
figures A, B, showing graphicjlly the results
of the study; preceded by an explanatory note
regarding their significance
Appendices:
A. The Cosmic Constant of Acceleration
B. Physical Dimensions of the Universe
C. Mathematics of the Cosmic Red Shift
D. Gravitational Fields of Earth, Sun,
Galaxy compared with the Cosmic Field
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SUMMARY
This paper presents the following formula for the cosmic red
shift, that is to say the red shift of 1 ight received from distant
galaxies:
Derivation of this formula is based on the hypothesis that
the cosmic red shift is a gravitational phenomenon and not a Doppler
effect, and it therefore interprets the cosmic red shift in terms of a
static rather than an expanding universe.
This hypothesis is, in turn, based on the concept of a cosmic
gravitational field whose strength is given by the acceleration
A photon of mass
2
g=.£. =
R
MG
R2
is pictured as losing energy progressively by interaction with this
cosmic field according to· the expression:
en~ = e fo = fn k = - -9. xL n- c2 ms fs
Inserting the expressions for g:
en fo L MG L
fs = - = - -., X
R RcL R
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It is to be noted that in Einstein's cosmology for his static model of
the universe, the expression:
MG
-2 = 1 Re
is a dimensionless constant of unit value, and that if it is written ,n
terms of the mass Ms and radius Rs of a star, it becomes the expression
for Einstein's gravitational red shift of 1 ight emitted from the surface
of a star. The relation between Einstein's gravitation~! red shift and
the cosmic red shift of the present hypothesis is discussed in the text.
Basis is shown in the text for the belief that the acceler-
2
ation g=L which measures the str.':.ngth of the cosmic gravitational field
R
has the same numerical value as Newton 1 s gravitational constant G:
g = ~
2
= :~ = 6.67x10-B cm sec
From these relations, the major physical dimensions of the
universe can be calculated, and are found to be:
R = 1 .35x 10 28 cm = 14.3x109 1 ight years
M = 1 .82x105 6 gms
V = 1 .25xJ085cm3
which are all acceptable values. We note also that for Einstein's static
model of the universe M=R2 numerically.
Newton's gravitational constant can now be inter~reted:
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which explains both the numerical value and the dimensions of G, and
shows that it is indeed an empirical constant. It is proposed that
= c 2 8 9 ;r- = 6.67xl0- cm.sec-2
be accepted as a hitherto unrecognized natural constant which measures
the strength of the cosmic gravitational field and which determines
the numerical value of Newton's co~stant Gas measured by laboratory
experiment.
From the data and figures given in the text, it will be seen
that the formula for the cosmic red shift derived from the present
hypothesis- ; ;.res red shift=distance relations which appear to be acceptc~~r;.
From the expression:
1 ight years
the djstance of a galaxy can be calculated from red shift data alone.
This should provide means for checking the validity of the present hy­pothesis,
thus sett! ing the question whether the universe is expanding
or static.
-1-
INTRODUCTfON
In the 1 ight of the recent detection of hitherto unsuspected
objects (quasi-ste11ar sources, or "quasars") seeming to 1 ie near the
limits of the observable universe, it is no.,., incumbent on science to
come to grips with the proper interpretation of the "red-shift" in the
spectra of extra-galactic objects. The apparently simplest interpre­tation
is that the red shift represents a Doppler effect, a true motion
of recession of the object away f--t' :::;-;: us, the observer, proport i ona 1 to
distance from us. Logical consequences of such an interpretation have
led to cosmologies involving an expanding--a dynamic--universe, with
philosophical imp1 ications that have been uncomfortable to many who
pondered the problem. Thus the Doppler effect has been accepted by
these scientists as only a provisional interpretation while other ap­proaches
have been sought. Recognizing that an alternative has been
the degradation of photon energy with distance and/or time, investig­ators
have devoted much apparently unsuccessful attention to such a
mechanism for energy loss. It is the object of the present paper to
suggest a possible mechanism and to explore its significance in relation
to certain "constants of nature", and its implications for cosmology--
a mechanism resulting from a study begun many years before developments
in the field of cosmology had given its outcome a measure of urgency.
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Over thirty years aso Tolman(3)gave an exhaustive discussion
of various possible models of the universe, from which we quote:
"The most unsatisfactory feature of the Einstein (static)
model as a basis for the cosmology of the actual universe is the finding
... .
that it provides no reason to expect any systematic shift in the wave
length of light from distant objects. In the actual universe, however,
the NOrk of Hubble and Humason shows a definite red shift which in­creases
at least very closely in 1 inear proportion to the distance. This
is of course the main consideration which will lead us to prefer non­static
to static models of the universe as a basis for actual cosmology.
c1~~ely connected with . this unsatisfactory feature of the
static Einstein model will be our later finding that the Einstein model
would not be stable."
These words still express the general views of cosmologists
regarding Einstein's original static model of the universe. The
principal purpose of this paper is to provide an explanation of the
red shift within Einstein's static model, supported by published
data regarding the red shift-distance-velocity relation. If this explan­ation
is correct, it is no longer necessary to rely upon the iheory of
the expanding universe to explain the red shift, and we may return to
Einstein's static model provided the question of stability can be
handled satisfactorily.
Extensive reading over a period of many years on the subject
of cosmological theory suggests the possibility that this supposed
(3) R. C. Tolman-11 Relativity, Thermodynamics anci Cosmology" (1934)
Reprinted 1946, 1949, 1958 .
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instability may not be in accord with the real nature of the universe.
It is based upon involved mathematics and tenuous assumptions, which
lead to the picture of bill ions of galaxies similar to our own, rushing
away from one another at speeds approaching the velocity of 1 ight. It
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would seem more realistic, and perhaps it is now timely, to assume
that after an existence of bill ions of years the universe has achieved
stability, or at least a quasi-stability, which permits us to think of
it in . terms of Einstein's static model without serio~s error.
The mathematics of the hypothesis presented here is based
upon Einstein's cosmo?ogy, and it yields results which provide an
acceptable red shift-distance ;elation without assuming a Doppler effect
due to recessional velocities. Also presented is an explanation of
Newton's gravitational constant in cosmic terms, as well as other in­teresting
and useful by-products. One such is a means of measuring
intergalactic distances unambiguously from red shift data alone, and
this relation offers means for testing the validity of the hypothesis
presented here. Another by-product is the concept of a general gravita­tional
field of the universe, something which seems never to be discussed
by cosmologists. Associated with this concept is the postulate that
there exists a hitherto unrecognized natural constant which measures the
strength of the cosmic gravitational field; which gives to Newton's
constant its nume~ical value; and which throws 1 ight upon Mach's famous
principle that the inertial properties of matter are due to the influence
of all of the matter of the universe.
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The cosmic field is pictured as the summation in space o~
the individual gravitational fields associated with all of the matter
contained in the universe; it must therefore be homogeneous and
isotropic in its gross properties, and it must permeate all space and
ponderable matter alike, without let or hindrance. Thus the gravitational
behavior and inertial properties of ponderable bodies are due to the inter­action
of their individual gravitational fields with the cosmic field.
In the last analysis the gravitational field associated with
each ponderable body is the summation in space of the gravitational
fields of the atoms of which it is composed, and the cosmic field r~
therefore the sunmation in space of the gravitational fields of all
of the atoms contained in the cosmos. Since atoms are also the
source of radiation (1 ight) and since radiation and gravitation are
the two known natural phenomena which pervade all space, one
suspects that the cosmic field may act as the transmitting agent, the
carrier, of radiation as it traverses space. This verges upon
metaphysical speculation, but perhaps after all it may be true . One
thinks of the classical concept of the aetheG and substitutes fo, it
the cosmic gravitational field, which as will be seen has definite
physical meaning.
Having thus prepared the reader for a radical departure from
previous cosmological views, we proceed to the argument.
UNITS- - - -
DIMENSIONS -
H
m
R = TT R
- C 2
G = 6.67xl o-8
g = 6.67xlo-8
C
V
L
t = L
C
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GLOSSARY OF SYMBOLS USED
The c.g.s. s.ystem is used throughout.
Where necessary, dimensions are indicated
by La Mb re
~s the total mass of the universe unless
otherwise indicated.
is sometimes used for the mass of a star.
is used for the mass of a relatively small
pondera~le body, or of a photon.
is sometimes used for the radius of a star.
is the radius of curvature of the universe
is the geode~ic radius corresponding to Rc.
IL3 M-1 r-21 is Newton's gravitational
constant.
l L r-~I is a postulated natura 1
cosmic acceleration.
is the velocity of light
is a velocity less than c
is a distance
constant of
is the relativistic time corresponding to the
distance L
represents a wave length. The subscripts (s)
and (o) represent wave lengths at source and
observer respectively.
~~=)-~represents a difference in wave length.
/\, s ~
f = c is a frequency corresponding to l
A
h
s,o
V B=c
is Plancks constant of action
subscripts sand o refer to properties
of photon at source and as observed
' represents velocity expressed as a fraction
of the velocity of I ight.
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GRAVITATION AND THE RED SHIFT
Let us assume that the red shift observed in light received
from dtstant galaxies is not a Doppler effect due to recessional
velocities but is a gravitational phenomenon; that is, a photon travers­ing
intergalactic space along its geodesic path at the velocity of light,
progressively loses energy by interaction between its "package" of
electromagnetrc energy and the isotropic general gravitational field
of the universe, which we shall speak of as "the cosmic field." We
can relate this red shift, which we shall call 11 the cosmic red shift",
to Einste , ,, 's fai,1il iar gravitational red shift of I ight radiated from- ·
intense gravitational fields cf cosmic bodies such as massive, dense
stars .
(1)
Very simply stated, Einstein based his prediction of this
gravitational red shift on the supposed effect of the gravitational field
upon the fundamental unit of time. According to this concept, an intense
gravitational field slows down the internal processes of the atom from
which the photon is emitted (or appears to do so), with the result that
the gravitational red shift has already taken place when the photon is
emitted.
(2)
Born likened this gravitational red shift to a fictive
Doppler effect, and showed that the red shift is equivalent to that which
would be measured by an observer having a recessional velocity relative
to the star equal to the escape velocity corresponding to the strength of
(1) EINSTEIN - 11 The Meaning of Relativity", Third Edition (1950) - P92
(2) BORN - 11 Einstein 1 s Theory of Relativity" (1962) - P35lff .
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the gravitational field at the surface of the star. Here the red shift takes
place not at the source but at the point of observation.
Using the principle of equivalence, Born also treated the gravita­tional
red shift as a Newtonian-I ike gravitational phenomenon, and showed
that the loss of energy by the photon (corresponding to its decreased fre­quency
or increased wave length) is precisely the gravitational poter.tial
energy of the photon (treated as a ponderable body) witr. respect to the
gravitational field of the star. His argument follows:
Light of frequency f can be regarded according to quantum theory as
cc~~isting of quanta of energy
These have an inertial mass
E = hf
m
_ E _ hf - - - -r c2 C
and this, according to the principle of equivalence, is equal to its gravita­tional
mass. When I ight quanta have traveled the distance L against the accel­eration
of gravitation, 11 a 11
, their energy has · decreased by mal. Hence at the end
of this journey the energy of a quantum E0 = hfo is only
hf0 = hfs - aLhfs = hfs (1-a~ ) V- C
If the factor his omitted from both sides, we obtain:
fo = fs(l- al)
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where subscripts sand o refer to conditions at the source and as observed.
Since the acceleration 14 a 11 of the gravitational field of a star is
given by:
we have
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Then since the mass of a photon is proportional to its frequency:
MsG
mo= ms - ms Rs2c2
ms-om=/:::;,. m = msx MsG L = ms MsG x ~
R/c2 Rsc2 Rs
and setting L = R(still following Born):
MsG
= R? s
This is the formula in general u5e for the gravitational red shift cf 1 ight
from a star.
If instead of fol lowing Bor o.'s procedure we take into account the
potential gradient of th'e ·-gravitational field of the star·, we arrive ·at the
same rretbematical result:
I, mc2 = -(~s MsG dl --
....... ·-R ['r
R c2 s
Thus if Einstein's gravitational re<l shift is interpreted in terms
of a progressive loss of energy by a photon, caused by interaction between
its packet of electromagnetic energy and the gravitational field of a star
as the photon "escapes" from the gravitational field, the mathematical
expression for the gravitational red shift is equivalent to Eh,stein's and
Born's expressions, and the. numerical results are the same.
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This seems to imply that the 9,avitational red shift may octualiy be
due to interaction bet~een the photon and tne gravitatio~al field of the stor as
the photon "escapes" from t~e fie1d, rather than to the effect of the
gravitationa: fie:~ of the star upon the ir.ternal processes of the emitting
ato~. Sorr. ~;d not s~ate that this is actually the mechanism which is
responsible for Lne gravitational rec sh:ft; w~a: he did was to show that
if one calc~]ates the ;ravitational red shift on tne basis of such an assump­tion,
Lhe result is ;dentica] w:th ~instein's oreclction based on an entirely
c:fferent tneory of tr.e phenomenon. The res~:t tnen does not ailow us to choose
either mechar.;s~.
~e prO?OSe to s~ow that the cos~:c red shift can be explained by
actua : · y
assu~ing that a photon/reacts wlth the cosmic field in such a manner that
lt progiessiveiy loses energy as it traveises intergalactic space.
however, in the case 0f tne cosmic red shift, we are dealing not
with the gravitational field of a stai ano its effect upon 1 ight radiated
from it, but with the hypothetical effect of the cosmic field upon a photon
traversing intergalactic space. The quest:on tnerefore arises, just what do
we mean by the cosmic fielci; wh~t is its nature?
It seeQS obvious that if the totai matter of the universe is more
or less uniformiy c:stributec throughout the universe, then intergalactic
space QUSt be occupied and permeated by the su~rnatlon in space of the gravi­tational
f:elds associated witn all of the matter in the universe from atoms
to galaxies, as well as inter5alacJic dust and gases. This field clearly
must be isotroplc, homogeneous, anc all-pervading, and it must have some
property by means of which its strengtn or intensity can be measured. Let
us see~ this property through the reo shift-distance relation of Hubble.
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This relation is given in many different ways, but the general
practice .has become, in recent years, to speak of velocity in miles or kilo­meters
per second per unit of distance - usually per millions of 1 ight years
or parsecs . If in relativistic fashion we express distar.ce as the time
required for 1 ight to traverse that distance, we may write:
.'l - .'l_ = constant (sec-1) L - ct
V V - = constant x c = - x c = a t L
where a has the dimensions of acceleration.
Einstein used the following expression for Hubble's constant as
the reciprocal of a distance related to the radius of the ~niverse;
H=.'lx 1
t c2 = ·Y., X l
C Ct
= .'l x 1 (cm-1)
C L
( See previous reference "Einstein" Footnote (1) Page 6.)
Hubble's constant Has given by Einstein is usually taken as the
reciprocal of the distance to the horizon of the universe, which is the
geodesic R. (See Appendix A). We therefore write from Einstein's equation:
.'l = Hc2 = c2 = 9
t R
where g is our postulated property, the acceleration which measures the
strength of the cosmic field.
Now if His the total mass of the universe, and R is its geodesic
radius, dimensional analysis lets us write
c2
R
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It is tempting to ascribe physical significance to this identific­atlon.
as it allows a relation between the Newtonian constant of gravitation,
G, and the cosmic constant of acceleration g. Support is given to this step
if the two right members are multiplied by ·MxR giving
a relation appearing in Einstein's model for a static universe, equating the
gravitational potential energy of the universe to its rest energy . . .
Since Hubble's red shift-distance linear relation was fir~t announced,
the yardstick for measuring cosmic distances has been revised several times,
always in the sense that distances are greater than had been supposed . The
result is that the cosmic acceleration corresponding to published data has
decreased steadily for . the past 30 years. In an appendix this is treated in
detail, but here we will give four such data points:
Year
1934
1945
1961
1963
(a)
v (km/sec)
Author L(1061 y)
(3) Tolman 172
(1) Einstein 133
(4) Shapley 32.2
(6) Gr·eenstein
(a)
22.2
Using the relativistic formula for B = Y = function
C
See previous reference "Max Sorn" Footnote (2) Page 6 .
a= y = V X C
t '['
54 .5x 10-8 cm/ sec 2
42 . 1 II
10.2 II
7.03 ti
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This tabulation suggests the possibility that over this period
of some 30 years the decreasiilg value of the acceleration "a", calculated
from published data, has approached as a lower limit the value of
6.67x10-Bcm/sec 2 , that is, numerically equal to the laboratory value of G, ...
the Newtonian constant. Let us assume this actually to be the case and
explore the ideas it presents.
From _q. =
G
it appears' that numerically M=R2. From g = f, the geodesic· radius of the
universe is 1 .35xto28cm (14.3xt09 1 ight-years). Thus M(=R2) is 1 .82x1056
grams; the volume of the universe is 1.25x108~-'-; 1 ) ; anc the mean density is
1 .46xto-29gm/cm3, all reasonable values.
The present hypothesis regarding the gravitational nature of the red
shift was initially developed without making use of Einstein's gravitational
red shift, ::>ecause the present hypoth,·sis is not primarily concerned wi:h the
concept of the effect of a gravitational field upon the fundamental unit of time,
nor with the internal processes of an atom. But when the mathematical expressions
·relating to the present hypothesis were worked out, they bore such a striking
resemblance to the expressions for the gravitational red shift that a connection
was indicated. This connection was found in Born's treatment of the gravita­tion.
al red shift, and we view this as strong support for the validity of the
< concept that the cosmic red shift is indeed a gravitational phenomenon, and
that the present hypothesis has a soL•rid theoretical basis.
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It is of course true that the isotropic homogeneous cosmic field is
a very different thing from the radial gravitational field of a star with its
inverse square potential gradient, but if a photon interacts with a stellar
gra~itationat field it is reasonable to suppose that it will also interact
with the cosmic field, though the interaction will of course differ in kind
and ·i{.) degree .
In the case of the
f~r-- -6
smat 1t=-I\:, = 2x10 for our
gravitational red shift, the Joss of energy is so
sun)that the mass of the photon may be taken as
constant and the restraining force therefore varies inversely as the square
of the distance i.e. with the strength of the field, and the gravitational
potential energy therefore varies inversely as the distance.
In the case of t tL cosmic red shift the strength of the field is
constant, but the loss of mass by the photon must be taken into account
because in effect that is what we are trying to measure in terms of an
increase of wave 1 ength :
red shift:
However, -if the changing mass is neglected, we find for the cosmic
= M X G
R c2
X L
R
where L is the distance traveled by the photon, Mand Rare the mass and
radius ot the universe. If L~R, this becomes the expression for Einstein's
gravitational red shift~hen Mand Rare interpreted as the mass and radius
of the star. And if .,.., e now take, MG = 1, as it i:.. for Einstein's static
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model of the universe, we have Hubble's linear relation of red shift versus
distance, except that the ratio is related to?.o instead of{s.
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,Expressing the red shift in terms of As, we have:
If Lis very small in reiation to R, the relation is still apprc;>ximately linear,
~ut as L increases, deviation from the linear relation increases and can no longer
be ignored. At the distance L=R the photon has lost all of its energy and
has ceased to exist. It is of some interest to note that for the largest
red shift thus far reported, we find according to this formula:
= 2
2
L = - R
3
This is very interesting indeed, but we still" have to take into
acco\. int the effect of the progressive Joss of mass of the photon accc;rding
to the present hypothesis. The correct formula for the change in mass, as
developed in an aRpendixr turns out to be:
L -gxc2
where ms and mo are the mass of the photon at the source and as observed,
and
MG 2 g=fiL=r
is the acceleration which measures the strength of the cosmic field, as
discussed earlier.
Since the cosmic red shift is always expressed in terms of the
ratio of the change of wave length~ A to the wave length at the source
A. s , (instead of the observed wave length /\ o), we now express the
cosmic red shift as an exponential function:
. -1!)-
-L
!!!.Q = 'R' e
ms L p 1f
= e -1
One observes that the numerical value of g does not appear in the
simpJ ified final expression for the cosmic red shift. '!~vertheless it is the
numerical value of g (the strength of the cosmic fieid) which determines the rate
of loss o ~ energy, because the numerical value of R was determined by the relation:
c2
g = ;r- = 6.67xJo-8 C2 _ J 9J . .
R = 6.67xJ0-8 - J~.3x10 y
To test the validity of this fc:-:--.u Ja for the cos:ni.: red shift,
we now compare results calculated from it with a) the J inear relation b) the
relativistic formula derived from Born's treatment of the subject. The i inear
relation is of course:
= V
C
The relativistic
= B
expression is;
~02_ %;2
B = =
~ 0 2 + ·A_s2
as derived from Born:
function ( .!:. )
R
This formula apparently has not yet been correlated with distance,
so we take for this formula the values of B corresponding to the red shift calculated -
L
from R ratios by the present hypothesis .
The results are shown in the following tabulation
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TABLE
L L (. It )-1 t-o 2 - ) s2 Divergence froo:
.R . (light year) !l 02 + nsz L.inear Relation
or
6,. ~ or (f~f}i ~~} / V R
B = - Bx- x s . C L s B
o. 001 6 o. 001 001 0.0010 1. 000 +0.10% +O .10% 14.3x10
6 0.01 1.43xl g O.Olo'TO 0.0100 1. 000 +1 .00% + 1 .00%
0.05 715x10
9
0.0513 0.0500 1. 000 +2.6 % +2.6 %
0.10 1.43xl o9 0.1052 0.0997 0.997 +5 .2 % +5.5 %
0.20 2.86x10 0.2214 o. 197 0.985 +11 .1% +12 .4%
0.50 7.l5x10~ 0.6487 0.462 0.924 +30 % +40. 1%
1.00 14.3x10
9
I. 718 0,762 o. 762 +72 % +125 %
1.10 15.7x10 2.004 0.800 0.728 +82 % +ISO % ------ - - - - - - - - - - - - - - - - - -
0.80 1 I .L;xJ09 1.2255 0.662 0.827 +54 % + 85 %
It seems to be generally accepted by recent writers on the subject
of the expanding universe (especially as related to Quasars), that values of B
calculated from the red shift by the relativistic formula correspond more or
les-s to the distance as measured by the ratio b.. For example the red shift
R
~ ~ 2 _00 is usually spoken of as indicating a distance of 80% of the radius 'Xs =
of the universe. The above Table shows that the distances b. are almost identical
R
with values of Bout to about b. = 0.2, but beyond this distance the divergence
R
becomes substantial. This should provide means for testing the validity of tha
present hypothesis.
Both formulae give a continuously increasing positive rate of diver­gence
from the 1 inear relation and such a divergence, according to Sandage(4) is
evidence that the universe is finite and bounded. Up to distances of about 200
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millions of I ight years the two formulae give practically the same result. Beyond
this distance divergence from the linear relation becomes increasingly important,
and the relativistic formula gives an increasingly larger divergence as compared
with the present hypothesis.
(4) Sandage: 11 The Red Shift,11 Scientific American, Sept. 19_56
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If the present hypothesis is correct, the distance L is accurately
.determined by the red shift data alone, provided no important part of the shift
,s due to the gravitational red shift at the source, or some other cause such as
a real, peculiar velocity of the source. It would therefore seem possible to check
the validity of this formula against actual astronomical data, especially for
distances less than say 700 mill ions of light years, where the distance can be
( estimated with reasonable accuracy by astronomical procedures. From the formula
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(
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for the cosmic red shift according to the present hypothesis, we derive for the
distance:
where R = 14.3 bill ions of light years
---- ------------------------------------------------------------ - ··----------------
During the preparation of this paper several important by-products
developed, three of which are given herewith under identifications A, B, C.
A) The gravitational fields associated with cosmic bodies may be regarded
as mere singularities in spac~ in relation to the cosmic field. Since we have
assumed a numerical value for the strength of the cosmic field
-8
g = 6.67x10 2 cm/sec
it seems appropriate to give substance to this statement. The strength of the
earth's gravitational field at its surface is:
HG
a= ---2 = 981 cm/sec2 . R
Compared with this the cosmic field appears to be pitifully weak, yet, by calcula­tion
from available data, we find the distances at which the strength of the gravita­tional
fields of the Earth, the Sun, and our galaxy decrease to that of the cosmic
field:
(
-,~-
Earth - 480x106 miles (between the orbits of Jupiter and Saturn)
Sun 0.047 1 ight years (about 1/100 of distance to nearest star)
Galaxy assumed concentrated at its center-21 ,100 1 ight years (about 2/3 of
galactic orbital radius of the Sun) ·
This is a rather startling result, and it shows that compared with the cosmic field,
the gra.vitational fields of cosmic bodies are indeed mere singularid~s in space.
Detailed calculations are given in Appenaix D.
B) For three centuries Newton's gravitational constant G has remained a
scient~fic enigma: what is its real physical meaning; what is the reason for
its specific numerical value; what is the explanation of its stra~ge dimensions;
is it actually a fundamental constant or an empirical constant?
We have postulated that the numerical value of G is determined by the
strength of the cosmic field, and have sho,,.,., that its physical meaning and its
dimensions are both explained by expressing Gin terms of our cosmic . constant
of acceleration:
Thus we see that G really is an empirical constant, which has served
as a kind of disguise for the fundamental constant g. That this relation has
escaped notice is of course due to the fact that the existence of the constant
. R2 g seems not to have been suspected, coupled with the fact that the rat,o __
M
which explains the dimension of G, must have unit value. If one wished to express
G in terms of an acceleration, it was only possible to write:
G = axL 2
M
but the questions then arises, what acceleration, what distance and what mass?
These questions have now been answered, in cosmic terms.
-,J-Using
our expression for G we find that the strength "a" of the
gravitational field of a ponderable body of mass m at distance L is:
mG
a= L2 = g x
"The .strength of the gravitational field of a ponderable body of mass mat distance
•
L is the strength of the cosmic field g multiplied by the ratio of the mass m of
the body to the total mass M of the universe, and by the square of the ratio of
the radius of the universe R to the distance L."
T~is seems a very satisfying interpretation of the gravitational pro­perties
of ponderable matter, and of Mach's principle that the inertial proper­ties
of matter are determined by the influence of surrounding matter, that is to
·say, of (he total matter in the universe. For if we write for the universe the
equation:
derived from Einstein's equations for the radius of curvature of his static model
of the universe, we have for a body of mass m:
mMG --= R
mc 2
and jn terms of g: .
mM R2
Rx g x M = mc2
mgR = mc2
"The mass-energy of a ponderable body is equal to its gravitational
energy with respect to the entire universe.11
and dividing by c 2 , we have:
m
r
(
(
l
(
-20-
"The inertial mass of a ponderable body is given by its gravitational
.energy with respect to the entire universe, divided by the square of the
velocity of 1 ight." --- .-----------------------------------------------------------------
CONCLUSION
If the present hypothesis is found correct, it is evident
that many of the most difficult . problems of astronomy and cosmology will be
alleviated; the theory of the expanding universe becomes unnecessary and indeed
unacceptable, and we return to Einstein's original static or quasistatic model of
the universe. Such a state of affairs would !:·e welcomed by many investigators
who find the concept of bill ions of galaxies rushing away from one another at
speeds approaching the velocity of 1 ight intel_Jectually repugoant. Rejection of
the theory of the expanding universe carries with it rejection of the 11 steady state"
theory and the concept of the "big bang" supposed to · have started the expansion of
the universe, and even of the recent revival of the concept of an oscillating
universe. But it does not rule out the possibility of a quasi-static universe
which may expand or contract slightly and very slowly without altering its
major physical dimensions to an important extent. Perhaps most important of all,
it is no longer necessary to fit the time required for the development of cosmic
bodies, including galaxies, into a time scale based on the theory of the expand­ing
universe. For the universe is then ageless and galaxies are free to run
their course at leisure, as indeed they must.
DESCRIPTION OF FIGURES
Figures A and B show the red shift-distance relation for four cases: ·
a) The line~r relation
A_ CJ\ L
~ = R
b) Born's non-relativistic relation, in which the Doppler
effect is ascribed to recessional motion of both galaxies at equal vel­ocities
in opposite directions:
B a f ~) = 2 t?;s
~I\ B(each galaxy= -------
'A+h
0 S
c) Born's relativistic relation, in which the relativity
effect at high velocity is taken into account. Here alt of the motion is
ascribed to the distant galaxy, and the formula, which is the one now in
general use, is: ·
B = f~) =
d) The formula of the present hypothesis:
L
R = e -1
Apparently there is stilt some uncertainty regarding the relation
between Band i for cases (b) and (c), but it seems to be generally agreed
that the relation is approximately 1 inear. Curves (b) and (c) are plotted
on the assumption that B = j and the results seem to support this view.
Figures A and B both show that formulae b, c, d alt give a marked
positive deviation from the linear relation, and that in atf cases the
deviation increases progressively with distance. The relativistic formula
-22-
gives the greatest deviation, the formula of the present hypothesis gives
the least deviation, while Born's non-relativistic formula gives intermediate
results, but somewhat closer to those of the present hypothesis.
For case (b) the velocity B is the sum of the two equal and opposite
velocities, so that each galaxy has one-half of the velocity shown. If the
relativistic effect is applied to each of these velocities and added to the
velocity effect, Curve (b) will be shifted to the left of Curve (c), giving
a higher deviation from linear than Curve (c).
Figure· A shows clearly that on the scale used, the data for formulae
b, c, dare so close together that the three curves are indistinguishable until
the distance L = 0.20 is approached, According to the present hypothesis this
represents a ~istance of 2.6 bill ions of 1 ight years, an~ the red shift of 0.22
is approximately that of the galaxies of constellation Hydra. This was, until
recently, the greatest distance at which red shifts had been measured. One
conclusion to be drawn is that the formula of the pre$ent hypothesis does not
lead to absurd results.
Figure B shows equally clearly that at ·distances gr~~ter thank= 0.20
ti·,f'; three curves becor.ie clearly distinguishable, and the deviations are sc great
that it should be possible to determine which gives results best in accord with
other astronomical data.
Another point to be mentioned is that if it is assumed that a quasar
is moving at roughly 0.80 of the velocity of light, the correction for absolute
magnitude includes a factor of 0.80 for decrease in apparent luminosity due to
velocity alone. If the present hypothesis is correct this factor drops out, and
we have a major change in absolute magnitud~. This is another means of testing
the validity of the various formulae.
On the other hand, according to the present hypothesis a red shift of
0.80 corresponds to a distance of!:.= 1 .1 or more than the distance to the
"horizon'of the universe, whereas fhe relativistic formula gives the distance
as!:. = 0.80. This may be helpful in determining whether an important fraction
of ~he red shift is due to Einstein's gravitational red shift, or to an actual
recessional velocity large enough to reconcile the various data involved. It
is quite possible that these strange objects do indeed have some kind of special
velocity without disrupting the concept of a generally static universe.
Since curves b, c, d ~re practically indistinguishable out to distance
L lf = 0.20, it seems desireable to give the actual data used for plotting them:
(
(
(
(
(
(
(
-23-
Born (No~J\.1 at iv i st i c) Born(Relattvistic)
~ 'As
.b,.f. 1
8=2 ~-?-- Ao2-~/
L R
-1 B B -Ao2 +/ls2 =
'As
e )-o+As R
0 .02 0 .0202 .... 0 .02 0.0198 0.0198
0.04 O .0408 0 .04 0.0392 0.0392
0.06 0.0618 0.06 0.0583 0.0582
0.08 0.0833 0.08 O .0770 0.0769
0. JO 0. I 052 0 .10 0.0952 O .0950
0. J 2 0. I 275 0. 12 0.1132 0.1131
0 .14 0. I 503 0 .14 O .1308 0 .1305
O. I 6 O. 1735 0 .16 0 .1482 0.1471
0 .18 O .1972 O. I 8 O .1650 0 .1640
0 .20 0.2214 0 .20 0 .1828 O .1803
0.22 O .1980 O .1960
From this tabulation it is clear that although curves b, c, dare
practically indistinguishable in Fig. A ~ut to distance k = 0.20, nevertheless
the data are not identical. Since any desired number of significant figures
can be used, in keeping with the accuracy of red shift data with which they
are to be correlated, it should be possible to make comparisons which would
determine which of these formulae is most nearly in keeping with astronomical
data.
If the present hypothesis is correct, the distances are given
unambiguously from red shift measurements alone:
where R=14.3xto9 1 ight years.
Correlation between observed and absolute luminosity is then greatly
simplified, for if the universe is static there is no correction for attenu­ation
due to velocity, nor for the relativistic Doppler effect. The only
corrections required are therefore those for distance (now accurately known)
and the effect of the red shift on energy distribution in the spectrum. Thus
it seems probable that the validity of the present hypothesis can be tested
without relying upon data relating to extremely remote galaxies or other
cosmic objects such as Quasars, for which data are difficult to obtain and
difficult to interpret.
(
(
(
(
(
(
(
(
(
\
(
l
~
t-z
,.,
v
cs, -. cs,
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0 .06
0.04
0 .02
0
AA
As
FIGURE A
Red shift-distance-velocity
relations for distance
·o
X
~ =0.2 i:=0.2214
(AVERY)
,1- .
up to
L R = 0.20 0
8 = 0.20 X
X
CU RYES 8, C, D
LINEAR RELATION
AA L
AS =R
FORMULA USED
L AA (. -) ( D) X AVERY As =\eR -,
(8) 0 BORN 8=2 AA =f(L)
A.o+A.s R
( C) • BORN 8= >..a2 - As"-= f (!:..)
A.02 t- AS2 R
I ~ (AVERY) B=f{~) (BORN) i X
(
(
(
l
z.o
,.8
( .. 6
(
(
'
(
(
('
.4
( .2
;
·.o
'"\.8
·1.6
(
.J .4
6A
As
Red shift- distance-velocity i ! / relations for distance
up to
L
·R = I
B= I
I .. CURVE /
0
x
• (B) o / I I X i ii I I X
• 0 I
CURVE (C) / / /x
I IX i /x/cuRVE (D) . 0/
/ Ix . I I Ix
-~ It . !J
Ill • 0
LINEAR RELATION
6A L
AS =R
FORMULA USED // ~~ .(I
I!
j~o
•or,
j;
~
~ (AVERY) ~=f(B) (BORN)
o..__...___...___.J..-_...1..-_..1..-_.J..-_..1..-_..1..-_......___......___......___......___~-~
O . 0 . 2 0 . 4 0 .6 0 .8 1.0 1.2 1.4

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Full Text

GRAV lT/; TI ON
and
r
THE RED SHIFT
Julian M. Avery
Apr i I , l 9E,6
Revised July, 1966
REFERENCES
(1) Einstein "The Meaning of Relativity"
Third Edition (1950)
(2) Max Born "Einstein's Theory of Relativity" (1962)
(3) R. C. Tolman - "Relativity, Thermodynamics and
Cosmo I ogy" (1934)
(Reprinted 1946, 1949, 1958)
(4) Harlow Shapley "Galaxies"
Revised Edition (1961)
(5) Sandage "The Red Shift"
Scientific American, Sept. 1956
(6) Greenstein "Quasi-Stellar Radio Sources"
Scientific American, Dec. 1963
(7) Sir Arthur Eddington "The Expanding Universe"
1st Edition 1933, 4th Impression 1946
t
l
TABLE OF CONTENTS
The format used in this paper is to provide first an
introduction to prepare the reader for what is to
follow; a glossary of Terms used in the Text; a Surrrnary
which gives briefly the findings which are presented;
a Text which develops the argument in a logical sequence
with the conclusions reached thereby; a set of Figures
showing graphically the results of the study, together
with a descriptive explanation; and a series of Appen­dices
which support the argument of the Text with appro­priate
detail. The reader may find it helpful to glance
at the Figures before reading the Text, in order that he
may have a clear idea as to the results the Text is lead­ing
up to as it proceeds.
i ntroduction
Glossary of Symbols Used in the Paper
The Text - Gravitation and The Red Shift
figures A, B, showing graphicjlly the results
of the study; preceded by an explanatory note
regarding their significance
Appendices:
A. The Cosmic Constant of Acceleration
B. Physical Dimensions of the Universe
C. Mathematics of the Cosmic Red Shift
D. Gravitational Fields of Earth, Sun,
Galaxy compared with the Cosmic Field
5
6
24
34 ·
37
43
(
!
(
(
(
(
(
(
ecause the present hypoth,·sis is not primarily concerned wi:h the
concept of the effect of a gravitational field upon the fundamental unit of time,
nor with the internal processes of an atom. But when the mathematical expressions
·relating to the present hypothesis were worked out, they bore such a striking
resemblance to the expressions for the gravitational red shift that a connection
was indicated. This connection was found in Born's treatment of the gravita­tion.
al red shift, and we view this as strong support for the validity of the
< concept that the cosmic red shift is indeed a gravitational phenomenon, and
that the present hypothesis has a soL•rid theoretical basis.
-13-
It is of course true that the isotropic homogeneous cosmic field is
a very different thing from the radial gravitational field of a star with its
inverse square potential gradient, but if a photon interacts with a stellar
gra~itationat field it is reasonable to suppose that it will also interact
with the cosmic field, though the interaction will of course differ in kind
and ·i{.) degree .
In the case of the
f~r-- -6
smat 1t=-I\:, = 2x10 for our
gravitational red shift, the Joss of energy is so
sun)that the mass of the photon may be taken as
constant and the restraining force therefore varies inversely as the square
of the distance i.e. with the strength of the field, and the gravitational
potential energy therefore varies inversely as the distance.
In the case of t tL cosmic red shift the strength of the field is
constant, but the loss of mass by the photon must be taken into account
because in effect that is what we are trying to measure in terms of an
increase of wave 1 ength :
red shift:
However, -if the changing mass is neglected, we find for the cosmic
= M X G
R c2
X L
R
where L is the distance traveled by the photon, Mand Rare the mass and
radius ot the universe. If L~R, this becomes the expression for Einstein's
gravitational red shift~hen Mand Rare interpreted as the mass and radius
of the star. And if .,.., e now take, MG = 1, as it i:.. for Einstein's static
. . rc2
model of the universe, we have Hubble's linear relation of red shift versus
distance, except that the ratio is related to?.o instead of{s.
-14-
,Expressing the red shift in terms of As, we have:
If Lis very small in reiation to R, the relation is still apprc;>ximately linear,
~ut as L increases, deviation from the linear relation increases and can no longer
be ignored. At the distance L=R the photon has lost all of its energy and
has ceased to exist. It is of some interest to note that for the largest
red shift thus far reported, we find according to this formula:
= 2
2
L = - R
3
This is very interesting indeed, but we still" have to take into
acco\. int the effect of the progressive Joss of mass of the photon accc;rding
to the present hypothesis. The correct formula for the change in mass, as
developed in an aRpendixr turns out to be:
L -gxc2
where ms and mo are the mass of the photon at the source and as observed,
and
MG 2 g=fiL=r
is the acceleration which measures the strength of the cosmic field, as
discussed earlier.
Since the cosmic red shift is always expressed in terms of the
ratio of the change of wave length~ A to the wave length at the source
A. s , (instead of the observed wave length /\ o), we now express the
cosmic red shift as an exponential function:
. -1!)-
-L
!!!.Q = 'R' e
ms L p 1f
= e -1
One observes that the numerical value of g does not appear in the
simpJ ified final expression for the cosmic red shift. '!~vertheless it is the
numerical value of g (the strength of the cosmic fieid) which determines the rate
of loss o ~ energy, because the numerical value of R was determined by the relation:
c2
g = ;r- = 6.67xJo-8 C2 _ J 9J . .
R = 6.67xJ0-8 - J~.3x10 y
To test the validity of this fc:-:--.u Ja for the cos:ni.: red shift,
we now compare results calculated from it with a) the J inear relation b) the
relativistic formula derived from Born's treatment of the subject. The i inear
relation is of course:
= V
C
The relativistic
= B
expression is;
~02_ %;2
B = =
~ 0 2 + ·A_s2
as derived from Born:
function ( .!:. )
R
This formula apparently has not yet been correlated with distance,
so we take for this formula the values of B corresponding to the red shift calculated -
L
from R ratios by the present hypothesis .
The results are shown in the following tabulation
I
(
(
(
(
(
(
(
(
(
(
(
(
L
...
-lb-
TABLE
L L (. It )-1 t-o 2 - ) s2 Divergence froo:
.R . (light year) !l 02 + nsz L.inear Relation
or
6,. ~ or (f~f}i ~~} / V R
B = - Bx- x s . C L s B
o. 001 6 o. 001 001 0.0010 1. 000 +0.10% +O .10% 14.3x10
6 0.01 1.43xl g O.Olo'TO 0.0100 1. 000 +1 .00% + 1 .00%
0.05 715x10
9
0.0513 0.0500 1. 000 +2.6 % +2.6 %
0.10 1.43xl o9 0.1052 0.0997 0.997 +5 .2 % +5.5 %
0.20 2.86x10 0.2214 o. 197 0.985 +11 .1% +12 .4%
0.50 7.l5x10~ 0.6487 0.462 0.924 +30 % +40. 1%
1.00 14.3x10
9
I. 718 0,762 o. 762 +72 % +125 %
1.10 15.7x10 2.004 0.800 0.728 +82 % +ISO % ------ - - - - - - - - - - - - - - - - - -
0.80 1 I .L;xJ09 1.2255 0.662 0.827 +54 % + 85 %
It seems to be generally accepted by recent writers on the subject
of the expanding universe (especially as related to Quasars), that values of B
calculated from the red shift by the relativistic formula correspond more or
les-s to the distance as measured by the ratio b.. For example the red shift
R
~ ~ 2 _00 is usually spoken of as indicating a distance of 80% of the radius 'Xs =
of the universe. The above Table shows that the distances b. are almost identical
R
with values of Bout to about b. = 0.2, but beyond this distance the divergence
R
becomes substantial. This should provide means for testing the validity of tha
present hypothesis.
Both formulae give a continuously increasing positive rate of diver­gence
from the 1 inear relation and such a divergence, according to Sandage(4) is
evidence that the universe is finite and bounded. Up to distances of about 200
<
millions of I ight years the two formulae give practically the same result. Beyond
this distance divergence from the linear relation becomes increasingly important,
and the relativistic formula gives an increasingly larger divergence as compared
with the present hypothesis.
(4) Sandage: 11 The Red Shift,11 Scientific American, Sept. 19_56
r
-17-
If the present hypothesis is correct, the distance L is accurately
.determined by the red shift data alone, provided no important part of the shift
,s due to the gravitational red shift at the source, or some other cause such as
a real, peculiar velocity of the source. It would therefore seem possible to check
the validity of this formula against actual astronomical data, especially for
distances less than say 700 mill ions of light years, where the distance can be
( estimated with reasonable accuracy by astronomical procedures. From the formula
(
l
(
l
for the cosmic red shift according to the present hypothesis, we derive for the
distance:
where R = 14.3 bill ions of light years
---- ------------------------------------------------------------ - ··----------------
During the preparation of this paper several important by-products
developed, three of which are given herewith under identifications A, B, C.
A) The gravitational fields associated with cosmic bodies may be regarded
as mere singularities in spac~ in relation to the cosmic field. Since we have
assumed a numerical value for the strength of the cosmic field
-8
g = 6.67x10 2 cm/sec
it seems appropriate to give substance to this statement. The strength of the
earth's gravitational field at its surface is:
HG
a= ---2 = 981 cm/sec2 . R
Compared with this the cosmic field appears to be pitifully weak, yet, by calcula­tion
from available data, we find the distances at which the strength of the gravita­tional
fields of the Earth, the Sun, and our galaxy decrease to that of the cosmic
field:
(
-,~-
Earth - 480x106 miles (between the orbits of Jupiter and Saturn)
Sun 0.047 1 ight years (about 1/100 of distance to nearest star)
Galaxy assumed concentrated at its center-21 ,100 1 ight years (about 2/3 of
galactic orbital radius of the Sun) ·
This is a rather startling result, and it shows that compared with the cosmic field,
the gra.vitational fields of cosmic bodies are indeed mere singularid~s in space.
Detailed calculations are given in Appenaix D.
B) For three centuries Newton's gravitational constant G has remained a
scient~fic enigma: what is its real physical meaning; what is the reason for
its specific numerical value; what is the explanation of its stra~ge dimensions;
is it actually a fundamental constant or an empirical constant?
We have postulated that the numerical value of G is determined by the
strength of the cosmic field, and have sho,,.,., that its physical meaning and its
dimensions are both explained by expressing Gin terms of our cosmic . constant
of acceleration:
Thus we see that G really is an empirical constant, which has served
as a kind of disguise for the fundamental constant g. That this relation has
escaped notice is of course due to the fact that the existence of the constant
. R2 g seems not to have been suspected, coupled with the fact that the rat,o __
M
which explains the dimension of G, must have unit value. If one wished to express
G in terms of an acceleration, it was only possible to write:
G = axL 2
M
but the questions then arises, what acceleration, what distance and what mass?
These questions have now been answered, in cosmic terms.
-,J-Using
our expression for G we find that the strength "a" of the
gravitational field of a ponderable body of mass m at distance L is:
mG
a= L2 = g x
"The .strength of the gravitational field of a ponderable body of mass mat distance
•
L is the strength of the cosmic field g multiplied by the ratio of the mass m of
the body to the total mass M of the universe, and by the square of the ratio of
the radius of the universe R to the distance L."
T~is seems a very satisfying interpretation of the gravitational pro­perties
of ponderable matter, and of Mach's principle that the inertial proper­ties
of matter are determined by the influence of surrounding matter, that is to
·say, of (he total matter in the universe. For if we write for the universe the
equation:
derived from Einstein's equations for the radius of curvature of his static model
of the universe, we have for a body of mass m:
mMG --= R
mc 2
and jn terms of g: .
mM R2
Rx g x M = mc2
mgR = mc2
"The mass-energy of a ponderable body is equal to its gravitational
energy with respect to the entire universe.11
and dividing by c 2 , we have:
m
r
(
(
l
(
-20-
"The inertial mass of a ponderable body is given by its gravitational
.energy with respect to the entire universe, divided by the square of the
velocity of 1 ight." --- .-----------------------------------------------------------------
CONCLUSION
If the present hypothesis is found correct, it is evident
that many of the most difficult . problems of astronomy and cosmology will be
alleviated; the theory of the expanding universe becomes unnecessary and indeed
unacceptable, and we return to Einstein's original static or quasistatic model of
the universe. Such a state of affairs would !:·e welcomed by many investigators
who find the concept of bill ions of galaxies rushing away from one another at
speeds approaching the velocity of 1 ight intel_Jectually repugoant. Rejection of
the theory of the expanding universe carries with it rejection of the 11 steady state"
theory and the concept of the "big bang" supposed to · have started the expansion of
the universe, and even of the recent revival of the concept of an oscillating
universe. But it does not rule out the possibility of a quasi-static universe
which may expand or contract slightly and very slowly without altering its
major physical dimensions to an important extent. Perhaps most important of all,
it is no longer necessary to fit the time required for the development of cosmic
bodies, including galaxies, into a time scale based on the theory of the expand­ing
universe. For the universe is then ageless and galaxies are free to run
their course at leisure, as indeed they must.
DESCRIPTION OF FIGURES
Figures A and B show the red shift-distance relation for four cases: ·
a) The line~r relation
A_ CJ\ L
~ = R
b) Born's non-relativistic relation, in which the Doppler
effect is ascribed to recessional motion of both galaxies at equal vel­ocities
in opposite directions:
B a f ~) = 2 t?;s
~I\ B(each galaxy= -------
'A+h
0 S
c) Born's relativistic relation, in which the relativity
effect at high velocity is taken into account. Here alt of the motion is
ascribed to the distant galaxy, and the formula, which is the one now in
general use, is: ·
B = f~) =
d) The formula of the present hypothesis:
L
R = e -1
Apparently there is stilt some uncertainty regarding the relation
between Band i for cases (b) and (c), but it seems to be generally agreed
that the relation is approximately 1 inear. Curves (b) and (c) are plotted
on the assumption that B = j and the results seem to support this view.
Figures A and B both show that formulae b, c, d alt give a marked
positive deviation from the linear relation, and that in atf cases the
deviation increases progressively with distance. The relativistic formula
-22-
gives the greatest deviation, the formula of the present hypothesis gives
the least deviation, while Born's non-relativistic formula gives intermediate
results, but somewhat closer to those of the present hypothesis.
For case (b) the velocity B is the sum of the two equal and opposite
velocities, so that each galaxy has one-half of the velocity shown. If the
relativistic effect is applied to each of these velocities and added to the
velocity effect, Curve (b) will be shifted to the left of Curve (c), giving
a higher deviation from linear than Curve (c).
Figure· A shows clearly that on the scale used, the data for formulae
b, c, dare so close together that the three curves are indistinguishable until
the distance L = 0.20 is approached, According to the present hypothesis this
represents a ~istance of 2.6 bill ions of 1 ight years, an~ the red shift of 0.22
is approximately that of the galaxies of constellation Hydra. This was, until
recently, the greatest distance at which red shifts had been measured. One
conclusion to be drawn is that the formula of the pre$ent hypothesis does not
lead to absurd results.
Figure B shows equally clearly that at ·distances gr~~ter thank= 0.20
ti·,f'; three curves becor.ie clearly distinguishable, and the deviations are sc great
that it should be possible to determine which gives results best in accord with
other astronomical data.
Another point to be mentioned is that if it is assumed that a quasar
is moving at roughly 0.80 of the velocity of light, the correction for absolute
magnitude includes a factor of 0.80 for decrease in apparent luminosity due to
velocity alone. If the present hypothesis is correct this factor drops out, and
we have a major change in absolute magnitud~. This is another means of testing
the validity of the various formulae.
On the other hand, according to the present hypothesis a red shift of
0.80 corresponds to a distance of!:.= 1 .1 or more than the distance to the
"horizon'of the universe, whereas fhe relativistic formula gives the distance
as!:. = 0.80. This may be helpful in determining whether an important fraction
of ~he red shift is due to Einstein's gravitational red shift, or to an actual
recessional velocity large enough to reconcile the various data involved. It
is quite possible that these strange objects do indeed have some kind of special
velocity without disrupting the concept of a generally static universe.
Since curves b, c, d ~re practically indistinguishable out to distance
L lf = 0.20, it seems desireable to give the actual data used for plotting them:
(
(
(
(
(
(
(
-23-
Born (No~J\.1 at iv i st i c) Born(Relattvistic)
~ 'As
.b,.f. 1
8=2 ~-?-- Ao2-~/
L R
-1 B B -Ao2 +/ls2 =
'As
e )-o+As R
0 .02 0 .0202 .... 0 .02 0.0198 0.0198
0.04 O .0408 0 .04 0.0392 0.0392
0.06 0.0618 0.06 0.0583 0.0582
0.08 0.0833 0.08 O .0770 0.0769
0. JO 0. I 052 0 .10 0.0952 O .0950
0. J 2 0. I 275 0. 12 0.1132 0.1131
0 .14 0. I 503 0 .14 O .1308 0 .1305
O. I 6 O. 1735 0 .16 0 .1482 0.1471
0 .18 O .1972 O. I 8 O .1650 0 .1640
0 .20 0.2214 0 .20 0 .1828 O .1803
0.22 O .1980 O .1960
From this tabulation it is clear that although curves b, c, dare
practically indistinguishable in Fig. A ~ut to distance k = 0.20, nevertheless
the data are not identical. Since any desired number of significant figures
can be used, in keeping with the accuracy of red shift data with which they
are to be correlated, it should be possible to make comparisons which would
determine which of these formulae is most nearly in keeping with astronomical
data.
If the present hypothesis is correct, the distances are given
unambiguously from red shift measurements alone:
where R=14.3xto9 1 ight years.
Correlation between observed and absolute luminosity is then greatly
simplified, for if the universe is static there is no correction for attenu­ation
due to velocity, nor for the relativistic Doppler effect. The only
corrections required are therefore those for distance (now accurately known)
and the effect of the red shift on energy distribution in the spectrum. Thus
it seems probable that the validity of the present hypothesis can be tested
without relying upon data relating to extremely remote galaxies or other
cosmic objects such as Quasars, for which data are difficult to obtain and
difficult to interpret.
(
(
(
(
(
(
(
(
(
\
(
l
~
t-z
,.,
v
cs, -. cs,
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0 .06
0.04
0 .02
0
AA
As
FIGURE A
Red shift-distance-velocity
relations for distance
·o
X
~ =0.2 i:=0.2214
(AVERY)
,1- .
up to
L R = 0.20 0
8 = 0.20 X
X
CU RYES 8, C, D
LINEAR RELATION
AA L
AS =R
FORMULA USED
L AA (. -) ( D) X AVERY As =\eR -,
(8) 0 BORN 8=2 AA =f(L)
A.o+A.s R
( C) • BORN 8= >..a2 - As"-= f (!:..)
A.02 t- AS2 R
I ~ (AVERY) B=f{~) (BORN) i X
(
(
(
l
z.o
,.8
( .. 6
(
(
'
(
(
('
.4
( .2
;
·.o
'"\.8
·1.6
(
.J .4
6A
As
Red shift- distance-velocity i ! / relations for distance
up to
L
·R = I
B= I
I .. CURVE /
0
x
• (B) o / I I X i ii I I X
• 0 I
CURVE (C) / / /x
I IX i /x/cuRVE (D) . 0/
/ Ix . I I Ix
-~ It . !J
Ill • 0
LINEAR RELATION
6A L
AS =R
FORMULA USED // ~~ .(I
I!
j~o
•or,
j;
~
~ (AVERY) ~=f(B) (BORN)
o..__...___...___.J..-_...1..-_..1..-_.J..-_..1..-_..1..-_......___......___......___......___~-~
O . 0 . 2 0 . 4 0 .6 0 .8 1.0 1.2 1.4